| L(s) = 1 | + 2·3-s − 2·5-s − 4·7-s + 2·13-s − 4·15-s + 2·19-s − 8·21-s + 3·25-s − 2·27-s − 4·31-s + 8·35-s − 10·37-s + 4·39-s − 12·41-s + 4·43-s − 12·47-s − 2·49-s + 18·53-s + 4·57-s − 4·61-s − 4·65-s + 10·67-s + 12·71-s − 8·73-s + 6·75-s + 8·79-s − 81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 1.51·7-s + 0.554·13-s − 1.03·15-s + 0.458·19-s − 1.74·21-s + 3/5·25-s − 0.384·27-s − 0.718·31-s + 1.35·35-s − 1.64·37-s + 0.640·39-s − 1.87·41-s + 0.609·43-s − 1.75·47-s − 2/7·49-s + 2.47·53-s + 0.529·57-s − 0.512·61-s − 0.496·65-s + 1.22·67-s + 1.42·71-s − 0.936·73-s + 0.692·75-s + 0.900·79-s − 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36966400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 96 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 18 T + 184 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 156 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 310 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T - 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.924247376909708048608849091088, −7.66783829725724784953336858756, −7.07741174382155122520269290921, −6.91628116394954160365956403857, −6.63248616674193658184615538611, −6.29163755502294297993715885201, −5.65785816604534465772795249599, −5.48710959631240336338778926536, −4.92097374666959412051852664861, −4.65415307504556607957279368121, −3.85170640384239659159690799558, −3.67026438125128953466856236138, −3.40336892870421047200662373514, −3.26314752845944657119478679825, −2.54234491275235465966533998415, −2.41359577446850311909584110893, −1.59581996368890417464949360355, −1.10485307753586601175468330643, 0, 0,
1.10485307753586601175468330643, 1.59581996368890417464949360355, 2.41359577446850311909584110893, 2.54234491275235465966533998415, 3.26314752845944657119478679825, 3.40336892870421047200662373514, 3.67026438125128953466856236138, 3.85170640384239659159690799558, 4.65415307504556607957279368121, 4.92097374666959412051852664861, 5.48710959631240336338778926536, 5.65785816604534465772795249599, 6.29163755502294297993715885201, 6.63248616674193658184615538611, 6.91628116394954160365956403857, 7.07741174382155122520269290921, 7.66783829725724784953336858756, 7.924247376909708048608849091088
